@DylanWilson isn't HKR already topological?

Let C be a cocomplete infinity category and Spc the infinity category of spaces. It appears to me that the functor colim: Spc_{/C} -> C is cocontinuous. Is there a reference and/or slick proof and/or good way of thinking about this?

What does 'Spc_{/C}' mean? What functor from C to Spc are you using for this overcategory? Even if you had one im

not sure what 'colim' would mean then

@DylanWilson I think he means spaces over C, that is Spaces×_{Cat} Cat_{/C}

I mean the category of pairs (X, F) with X a space and F \in Fun(X, C). This is the same as the slice category Spc_{/C'} [is this the right notation?], where C' is the core groupoid of C. [I think there may be some size issues.]

(i.e. what @DenisNardin said)

Ah! Got it.

I think it is the left adjoint of the functor sending c to the core of C_{/c}

(I usually denote the core of C with iC but there are too many notations for the core...)

Sounds plausible. Thanks!

@DenisNardin What would I do without you ^^

@AaronMazel-Gee Thanks that's very interesting. Do you have anything written yet?

Is it correct that the E-infty structure on the cobar of a commutative Hopf algebroid can always be picked in such a way that [x] \cup_1 [y] = [xy] on the nose? Anyone know a reference?

(Where the RHS is also graded of course.)

Write $X$ as the geometric realization of an 'S-free' simplicial spectrum, i.e. one obtained by attaching wedges of spheres at each latching-matching step. Applying $\pi_*$ and taking normalized chains will give a $\pi_*$-free resolution of $\pi_*X$. On the other hand, map the simplicial object into $Y$ and note that a map of a wedge of spheres $W$ into $Y$ is the same as a $\pi_*$-modules map of $\pi_*W$ to $\pi_*Y$ for silly Yoneda-like reasons. Now you've got a cosimplicial gadget and that

always gets you a spectral sequence (with a priori horrible convergence) whose E_2-term you can compute by applying $\pi_*$ everywhere then computing cohomology. By the remark at the beginning, you see this cochain complex computes Ext.

And then you have to contend with convergence

Ok, Dylan was faster :P

but it turns out okay in this case. And yes, as Achim says- it works for any at least E_1-ring spectrum... I guess I'm not so sure about arbitrary ring spectra, since there's not so great of a 'htpy theory of E-modules' in this case.

@DylanWilson Sorry but I don't think I understand how you want to filter $X$? What do you mean by "latching-matching" step?

@SaalHardali Here is a(n equivalent) construction of the SS: take π_*X. This is a π_*S-module, and let \{x_i\}_{i\in I_0} be a set of generators as a π_*S-module. Let F_0=⨁_{i∊I_0}S . You get a map F_0→X. Let the fiber be K_0. Then iterate the procedure, so that you get X→ΣK_0→Σ²K_1→... . Then take [,Y] and you get an exact couple, which gives you a SS with E_2 term as promised.

@DylanWilson Let me rephrase your answer to see if i understand. As soon as you express $X$ as a filtered colimit of free guys then one gets a free resolution of $\pi_* X$ by taking $\pi_*$ and then normalized chains. Then by applying $Y(-)$ spectral sequence which wants to converge to $[X,Y]_*$.

Aha so its a filtered limit!

Nevermind filtered

Its a limit.

Filtered? I think that Dylan was thinking of the simplicial version (there is a stable Dold-Kan correspondence that shifts from a filtered thing to a (co)simplicial thing)

Yah I was mixing stuff up.

Anyway the spectral sequence comes from expressing $X$ as a limit right? not a colimit.

So, I don't want to read Dylan's mind here, but I'm pretty sure he meant colimit.

That makes more sense.

Either way, a reference is EKMM theorem IV.4.1

But then i'm confused by the remark that normalized chains of the simplicial diagram gives a reslution of $\pi_* X$.

For some reason there they call "universal coefficient spectral sequence" only a special case, but I think everyone uses the name for the general case as well (ditto for "Künneth spectral sequence")

What is EKMM?

Elmendorf, Kriz, Mandell, May - Rings modules and algebras in stable homotopy theory

In your explicit construction the filtration can be defined as $F_k X = fib (X \to \dots \to \Sigma^k K_{k-1}) $ right?

I think, I don't remember exactly how the stable Dold-Thom goes. It's not super important, what counts is the exact couple

Let $\mathcal{V}$ be a monoidal $\infty$-category. Does anyone know whether the $\infty$-category of $\infty$-categories enriched in $\mathcal{V}$ is an $\infty$-cosmos (as developed by Riehl and Verity)? If not in general, does it become true if I add appropriate adjectives to $\mathcal{V}$ such as Cartesian?

@DylanWilson What did you mean when you said that "taking $\pi_*$ gives a free resolution of $\p_*X$"? At best I got an exact couple...

@SaalHardali Taking $π_*$ you get a simplicial $π_*S$-module, which under Dold-Kan becomes a complex of $π_*S$-modules which happens to be a free resolution of $π_*X$

1. I don't think it does, in general; 2. I'll leave Dylan to answer that because I can never keep the story straight on the simplicial point of view, that's why I prefer the approach with the filtration

In terms of filtrations how do i get the story right?

I mean, by the steps you outlined earlier I only manage to get an exact couple

So, let me write a little more details. We have a bunch of fiber sequences K_i→F_i→K_{i-1} where K_{-1}=X, F_i is a wedge of spheres and π_*F_i→π_*K_{i-1} is surjective. Are you ok with this?

So the les of the fiber sequences degenerate and we get ses 0→π_*K_i→π_*F_i→π_*K_{i-1}→0

In particular, by splicing those together, we get a les 0←π_*X←π_*F_0←π_*F_1←..., that is a free π_*S-module resolution of π_*X

Instead of applying π_* now apply [-,Y]_*. You get an exact couple, whose E_1 page is Hom_{π_*S}(π_*F_i, π_*Y) and the differential is given by the differential in the above resolution

(this is because if F is a wedge of spheres [F,Y]_* = Hom_{π_*S}(π_*F,π_*Y) )

But this tells us that the E_2-page is the Ext you were asking about before!

Perfect! Thanks!

Bonus: if instead you apply π_*(-∧Y) you get the Künneth spectral sequence

Is it correct to say that all adams-type spectral sequences are special cases of this where instead we are in some module/comodule category?

For an example of what i mean the classical cohomological adams spectral sequence is this construction in the category of modules over $End_{\mathbb{S}}(H\mathbb{F}_2)$

@CharlesRezk yes it's all written up, just making some final tweaks. if you'd be interested in reading the paper intros i can send you those, or alternatively it should all appear on the arxiv within the next week or so

@AdrianClough first of all, i like the moomin! also, i thought the RV stuff is based on categories enriched over a suitably nice model 1-category? perhaps related to your underlying hope though, note that for any monoidal $V$, the yoneda embedding $V \to P(V) := Fun(V^{op},S)$ is monoidal for day convolution on the target, which crucially becomes presentably monoidal (so basically all things you want to be true are true).

(and since the yoneda embedding is an embedding, it's a condition for a $P(V)$-enriched $\infty$-category to actually be $V$-enriched -- in other words, they contribute a full subcategory.)

@SaalHardali the simplicial thing lets you easily see what the E_2 term is, which is why I like it. As for the 'associated filtration': in general, if you have a simplicial widget, and a homological functor H for widgets, then you get a spectral sequence computing H of the geometric realization in terms of H of the pieces (the E_1 term is H on each piece, and even better, the d_1 differential is the alternating sum of the face maps!). This is due to Segal

(classifying spaces and spectral sequences)

the filtration comes from the skeletal filtration

i.e. this agrees with the spectral sequence associated to the skeletal filtration on the geometric realization (whcih will look like what Denis described).

as for the latching-matching blah blah thing, there's lots of references but one such is HA.7.2.1

@SaalHardali No, Adams spectral sequences are not a special case of this construction.

they come not from approximating an E-module by free E-modules, but by 'approximating a spectrum by E-modules', but in a different sort of way.

the universal coefficient theorem for maps F(X,HF_2) -> HF_2 over the Steenrod-algebra-spectrum End(HF_2) is absolutely the Adams spectral sequence

@TylerLawson This is exactly what I meant, Thanks!

whoa- shouldn't have spoken so soon. sorry! and thanks tyler!

@CharlesRezk actually maybe i was being silly earlier, and we just need to consider the cyclotomic trace for the stable $\infty$-category of (cohomologically bounded complexes of) vector bundles on our space $X$. (i.e., there's no need to have a full algebro-geometric context for all of this.)

Is the functor $Spaces \to (E_{\infty}-Ring_{aug})^{op}$ given by Ind-extending the spanier whitehead dual functor on finite homotopy types an embedding?

Sorry the left hand side should be the Spanier whitehead category

And forget the Ind-extension just $X \mapsto \mathbb{D}(X)$

@AaronMazel-Gee Haha, thanks! I was brought up on Moomin :) Anyways, are you saying that I might get away with no conditions on monoidal $V$, because $P(V)$ already has all the nice properties I might hope for?

@AaronMazel-Gee Also, a fairly straightforward way of getting the underlying quasicategory-enriched category of the $\infty$-cosmos should simply come from Haugseng's paper on rectifying enriched $\infty$-categories: If $V$ is modelled by some model category $V'$, and the category of all categories enriched in $V'$ is itself a model category modelling all $V$-enriched $\infty$-categories, one could hope that this is canonically a model category enriched in the model category of quasicategories.

However I have only just started thinking about these things, so I have no idea if this ever happens, and if yes, whether the resulting quasicategory-enriched category of fibrant objects can be given the structure of an $\infty$-cosmos.